Truncated Levy Walk Research Articles

Analysis of Cell Sojourn Time in Heterogeneous Networks With Small Cells

Recently, heterogeneous networks with small cells have been widely used to increase the capacity of mobile systems. In such environments, accurate estimation of the mean cell sojourn time is critical for evaluating the performance of the network and its applications. It is especially important to analyze the cell sojourn times of mobile users as they reside in different network tiers: either macro-cell-only or small-cell-covered areas. But because small cells are deployed in an irregular manner, it is difficult to derive the analytical mean cell sojourn time in a macro-cell-only area. In this letter, we propose a novel approach to resolve this difficulty. We developed a simple but effective trick that approximates the heterogeneous network to a discrete grid so that it becomes tractable, making it possible to derive the analytical mean sojourn time in the macro-cell-only area. Simulation results confirm that the proposed method has excellent accuracy for general random walk mobility models including random waypoint, Brownian motion, tailored Brownian motion, and truncated Levy walk.

Performance Evaluation of Efficient Routing Protocols in Delay Tolerant Network under Different Human Mobility Models

Delay and/or Disruption-Tolerant Networking (DTN) is a novel communication prototype that can span across multiple networks and deal with unpredicted conditions in the Internet model.Delay-tolerant networks (DTNs) are partitioned wireless ad hoc networks with intermittent connectivity. Additional terminology in this family of dynamic networks includes disruption-tolerant networks, intermittently connected networks, and opportunistic networks. Routing of the packets in DTN is based on store-carry-and forward paradigm. In this paper, we study and analyze performance of well known PROPHET and Spray and Wait routing protocol, under different human mobility models such as Truncated Levy Walk mobility model (TLW),Self-similar Least Action Walk (SLAW) and Random way point (RWP) model. The MATLAB simulator is used in order to analyze the performance of these routing protocols.Simulation results illustrate that Spray and Wait significantly outperforms the PROPHET on aspects of delivery ratio, average delay and communication overhead.

Towards Understanding the Fundamentals of Mobility in Cellular Networks

Despite the central role of mobility in wireless networks, analytical study on its impact on network performance is notoriously difficult. This paper aims to address this gap by proposing a random waypoint (RWP) mobility model defined on the entire plane and applying it to analyze two key cellular network parameters: handover rate and sojourn time. We first analyze the stochastic properties of the proposed model and compare it to two other models: the classical RWP mobility model and a synthetic truncated Levy walk model which is constructed from real mobility trajectories. The comparison shows that the proposed RWP mobility model is more appropriate for the mobility simulation in emerging cellular networks, which have ever-smaller cells. Then we apply the proposed model to cellular networks under both deterministic (hexagonal) and random (Poisson) base station (BS) models. We present analytic expressions for both handover rate and sojourn time, which have the expected property that the handover rate is proportional to the square root of BS density. Compared to an actual BS distribution, we find that the Poisson-Voronoi model is about as accurate in terms of mobility evaluation as hexagonal model, though being more pessimistic in that it predicts a higher handover rate and lower sojourn time.

Normalized truncated Levy walk applied to flexible pavement performance

Government agencies and consulting companies face the challenge of pavement management aiming to maintain pavements in serviceable condition in the long term. This paper analyzed the evolution of the performance of flexible pavements to forecast the change in the serviceability level offered by the structure. Data from the Minnesota Road Research Project (MnROAD) road test corresponding to the low-volume traffic loop were used in the study. The data analysis centered on the normalized truncated Levy walk model. It was concluded that the truncated Levy distribution optimally describes the decrease of pavement serviceability caused by traffic and climatic conditions specific to the infrastructure location and traffic. The Truncated Levy Flight (TLF) provided a better approximation over the full range of the normalized pavement serviceability index than the Gaussian distribution for dataset considered.

Analytical representation of stock and stock-indexes returns: Non-Gaussian random walks with various jump laws

Empirical data for stock and stock-indexes returns that is available for international markets as well as for the Russian stock market is introduced and discussed. Random walk process with a specific law of an elementary independent increment (jump) in some random walk space is proposed for a proper basic description. An analytical representation of a random walk distribution law is derived from a general conception of stochastic elementary jumps. This representation covers classic S. Chandrasekhar’s results for Gauss random walks with limited jumps and jumps having distributions with all moments as well as P. Levy flights with jump distributions without moments starting from the second one. Finally, it describes truncated Levy random walks for jumps having distributions with at least the second moment. The natural generalization of truncated Levy walks is done. The random walk process is compared with known empirical data for the above discussed return fluctuations of international assets and stock indexes. There is a very good agreement of the elaborated theory with empirical data.

On the Levy-Walk Nature of Human Mobility

We report that human walk patterns contain statistically similar features observed in Levy walks. These features include heavy-tail flight and pause-time distributions and the super-diffusive nature of mobility. Human walks are not random walks, but it is surprising that the patterns of human walks and Levy walks contain some statistical similarity. Our study is based on 226 daily GPS traces collected from 101 volunteers in five different outdoor sites. The heavy-tail flight distribution of human mobility induces the super-diffusivity of travel, but up to 30 min to 1 h due to the boundary effect of people's daily movement, which is caused by the tendency of people to move within a predefined (also confined) area of daily activities. These tendencies are not captured in common mobility models such as random way point (RWP). To evaluate the impact of these tendencies on the performance of mobile networks, we construct a simple truncated Levy walk mobility (TLW) model that emulates the statistical features observed in our analysis and under which we measure the performance of routing protocols in delay-tolerant networks (DTNs) and mobile ad hoc networks (MANETs). The results indicate the following. Higher diffusivity induces shorter intercontact times in DTN and shorter path durations with higher success probability in MANET. The diffusivity of TLW is in between those of RWP and Brownian motion (BM). Therefore, the routing performance under RWP as commonly used in mobile network studies and tends to be overestimated for DTNs and underestimated for MANETs compared to the performance under TLW.

Analytical representations of non-Gaussian laws of random walks

An analytical representation of a random process with independent increments in some space (random walks introduced by Pearson) is considered. The law of random walk distribution in space is derived from the general representation of stochastic elementary hops (distribution law of hop probability) using Kadanoff’s concept of the unit increment as one hop. For limited hop laws and laws of hop distributions with all moments there naturally arises Chandrasekhar’s result that describes ordinary physical diffusion. For laws of hop distributions without the second and highest moments there also arise known Levy walks (flights) sometimes treated as superdiffusion. For the intermediate case, where the distributions of hops have at least the second moment and not all finite moments (these hops are sometimes called truncated Levy walks), the asymptotic form of the random walk distribution was obtained for the first time. The results obtained are compared with the experimental laws known in econophysics. Satisfactory agreement is observed between the developed theory and the empirical data for insufficiently studied truncated Levy walks.

Tsallis and Levy statistics in the preparation of an earthquake

Abstract. Precursory fracture induced electromagnetic (EM) emissions, rooted in opening cracks and ranging from MHz to kHz, with the MHz appearing earlier, are produced and detected both at laboratory and geophysical scale. Recently, we have proposed the following two epochs/stages model of EQ generation: (i) The final kHz part is triggered by the fracture of high strength and large asperities that are distributed along the activated fault and sustain the system. (ii) The initial MHz part is thought to be due to the fracture of highly heterogeneous system that surrounds the family of asperities. Interestingly, the MHz EM time-series can be described in analogy with a thermal second order phase transition. Herein we focus on the MHz pre-seismic activity, and especially on the naturally arising question: what is the physical mechanism that organizes the heterogeneous system in its critical state? Combining ideas of Levy and Tsallis statistics and criticality with features hidden in the precursory MHz time-series we argue that a Levy walk type mechanism can organize the heterogeneous system to criticality. Based on a numerically produced truncated Levy walk, we propose a way to estimate in the stage of critical fluctuations: (i) the associated Levy index-a, which describes quantitatively the underlying Levy dynamics, and (ii) the range of values where the nonextesitive Tsallis index q is restricted. We also show that the kHz EM activity could not be described by a truncated Levy mechanism. This result further indicates an abrupt sweep of the population of asperities that sustain the system.

Normalized truncated Levy walks applied to the study of financial indices

This work is devoted to the study of the statistical properties of financial indices from developed and emergent markets. We performed a new analysis of the behavior of several financial indices by using a normalized truncated Levy walk model. We conclude that the truncated Levy distribution describes perfectly the evolution of the financial indices near a crash for both well-developed and emergent markets.